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I'm trying to solve this question:

For a stochastic process to be a Weiner Process it must have these properties:

  1. $W(0) = 0$ so $W^2(0) = 0$

  2. $E(W(t)) = 0$ but $E(W^2(t)) = t$ I think this is enough to prove that $W^2(t)$ is not a Weiner process

But I also want to prove the following:

  1. $W^2(u) - W^2(t)$ is indepedent from $W^2(s) - W^2(r)$ given $u > t > s > r > 0$

In a Winer process the increments are normally distributed so to prove independence, I need to prove their covariance is $0$. But I don't know how to find if the increments for $W^2(t)$ are normally distributed or if there's another way to prove independence.

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Broken_Window
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  • The title of your question is somewhat misleading. – saz Mar 26 '16 at 07:01
  • As the linked question shows, $W^2(t)$ does not have independent increments. – saz Mar 26 '16 at 07:33
  • @saz I randomly found the question I linked, tried to do it myself and get stuck proving the independence of the increments. I didn't know that part was already asked facepalm. Thanks for the link! – Broken_Window Apr 01 '16 at 14:01

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